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Abstract: We study scale anomalies in conformal field theories. We first use Euclidean position space to show that scale anomaly coefficients can be computed in terms of CFT data (dimensions and OPE coefficients). We give an explicit expression in the form of a sum over operators, and we show that the series is not positive-definite. We then use a different approach by coupling the CFT to a weakly-coupled probe quantum field in Minkowski space. The scattering amplitudes of the probe particles are sensitive to the scale anomaly, and the optical theorem then gives an expression for the scale anomaly coefficient in terms of a positive sum of states.